A Harmonic Analysis Solution to the Basket Arbitrage Problemaspremon/PDF/INFORMS05harmonic.pdf · A...

A Harmonic Analysis Solution to the Basket

Arbitrage Problem

Alexandre d’Aspremont

ORFE, Princeton University.

A. d’Aspremont, INFORMS, San Francisco, Nov. 14 2005. 1

Introduction

• Classic Black & Scholes (1973) option pricing based on:

◦ a dynamic hedging argument◦ a model for the asset dynamics (geometric BM)

• Sensitive to liquidity, transaction costs, model risk ...

• What can we say about derivative prices with much weaker assumptions?

Static Arbitrage

Here, we rely on a minimal set of assumptions:

• no assumption on the asset distribution

• one period model

An arbitrage in this simple setting is a buy and hold strategy:

• form a portfolio at no cost today with a strictly positive payoff at maturity

• no trading involved between today and the option’s maturity

What for?

• Data validation (e.g. before calibration), static arbitrage means marketdata is incompatible with any dynamic model. . .

• Test extrapolation formulas

• In illiquid markets, find optimal static hedge

Outline

• Static Arbitrage

• Harmonic Analysis on Semigroups

• No Arbitrage Conditions

Simplest Example: Put Call Parity

payoff

Put Call−

= K − S

Static Arbitrage: Calls

Also, necessary and sufficient conditions on call prices:

Suppose we have a set of market prices for calls C(Ki) = pi, then there is noarbitrage iff there is a function C(K):

• C(K) positive

• C(K) decreasing

• C(K) convex

• C(Ki) = pi and C(0) = S

This is very easy to test. . .

80 85 90 95 100 105 110 115 1200

strike price

option

Dow Jones index call option prices on Mar. 17 2004, maturity Apr. 16 2004

Source: Reuters.

Data quality...

• All the prices are last quotes (not simultaneous)

• Low volume

• Some transaction costs

Problem: this data is used to calibrate models and price other derivatives...

Dimension n: Basket Options

• A basket call payoff is given by:

wiSi − K

where w1, . . . , wk are the basket’s weights and K is the option’s strikeprice

• Examples include: Index options, spread options, swaptions...

• Basket option prices are used to gather information on correlation

We denote by C(w, K) the price of such an option, can we get conditions totest basket price data?

Necessary Conditions

Similar to dimension one...

Suppose we have a set of market prices for calls C(wi, Ki) = pi, and there isno arbitrage, then the function C(w, K) satisfies:

• C(w, K) positive

• C(w, K) decreasing in K, increasing in w

• C(w, K) jointly convex in (w, K)

• C(wi, Ki) = pi and C(0) = S

This is still tractable in dimension n as a linear program.

Sufficient?

A key difference with dimension one: Bertsimas & Popescu (2002) show thatthe exact problem is NP-Hard.

• These conditions are only necessary...

• Numerical cost is minimal (small LP)

• We can show sufficiency in some particular cases

In practice: these conditions are far from being tight, how can we refinethem?

Arrow-Debreu prices

• Arrow-Debreu: There is no arbitrage in the static market iff there is aprobability measure π such that:

C(w, K) = Eπ(wTx − K)+

• π(x) represents Arrow-Debreu state prices.

• Discretize on a uniform grid: This turns this into a linear program with mn

variables, where n is the number of assets xi and m is the number of bins.

• Numerically: hopeless. . .

• Explicit conditions derived by Henkin & Shananin (1990) (link with Radontransform), but intractable. . .

Tractable Conditions

• Bochner’s theorem on the Fourier transform of positive measures:

f(s) =∫

e−i<s,x>g(x)dx with g(x) ≥ 0

f(s) positive semidefinite

which means testing if the matrices f(sisj) are positive semidefinite

• Can we generalize this result to other transforms? In particular:

(wTx − K)+dπ(x)

Outline

Harmonic Analysis on Semigroups

Some quick definitions...

• A pair (S, ·) is called a semigroup iff:

◦ if s, t ∈ S then s · t is also in S

◦ there is a neutral element e ∈ S such that e · s = s for all s ∈ S

• The dual S∗ of S is the set of semicharacters, i.e. applications χ : S → R

such that

◦ χ(s)χ(t) = χ(s · t) for all s, t ∈ S

◦ χ(e) = 1, where e is the neutral element in S

• A function f : S → R is positive semidefinite iff for every family {si} ⊂ S

the matrix with elements f(si · sj) is positive semidefinite

Harmonic Analysis on Semigroups

Last definitions (honest)...

• A function α is called an absolute value on S iff

◦ α(e) = 1◦ α(s · t) ≤ α(s)α(t), for all s, t ∈ S

• A function f is bounded with respect to the absolute value α iff there is aconstant C > 0 such that

|f(s)| ≤ Cα(s), s ∈ S

• f is exponentially bounded iff it is bounded with respect to an absolutevalue

Carleman type conditions on growth for moment determinacy, etc. . .

Harmonic Analysis on Semigroups: Central Result

The central result, see Berg, Christensen & Ressel (1984) based onChoquet’s theorem:

• the set of exponentially bounded positive definite functions is a Bauersimplex whose extreme points are the bounded semicharacters...

• this means that we have the following representation for positive definitefunctions on S:

f(s) =

χ(s)dµ(χ)

where µ is a Radon measure on S∗

Harmonic Analysis on Semigroups: Simple Examples

• Berstein’s theorem for the Laplace transform

S = (R+,+), χx(t) = e−xt and f(t) =

e−xtdµ(x)

• with involution, Bochner’s theorem for the Fourier transform

S = (R,+), χx(t) = e2πixt and f(t) =

e2πixtdµ(x)

• Hamburger’s solution to the unidimensional moment problem

S = (N,+), χx(k) = xk and f(k) =

xkdµ(x)

Outline

The Option Pricing Problem Revisited

What is the appropriate semigroup here?

• Basket option payoffs (wTx − K)+ are not ideal in this setting.

• Solution: use straddles: |wTx − K|

• Straddles are just the sum of a call and a put, their price can be computedfrom that of the corresponding call and forward by call-put parity.

• The fact that |wTx − K|2 is a polynomial keeps the complexity low.

Payoff Semigroup

• The fundamental semigroup S here is the multiplicative payoff semigroupgenerated by the cash, the forwards and the straddles:

S = {1, x1, . . . , xn, |wT1 x − K1|, . . . , |w

Tmx − Km|, x2

1, x1x2, . . .}

• The semicharacters are the functions χx : S → R which evaluate thepayoffs at a certain point x

χx(s) = s(x), for all s ∈ S

The Option Pricing Problem Revisited

• The original static arbitrage problem can be reformulated as

find f

subject to f(|wTi x − Ki|) = pi, i = 1, . . . , m

f(s) = Eπ[s], s ∈ S (f moment function)

• The variable is now f : S → R, a function that associates to each payoff s

in S, its price f(s)

• The representation result in Berg et al. (1984) shows when a (price)function f : S → R can be represented as

f(s) = Eπ[s]

Option Pricing: Main Theorem

If we assume that the asset distribution has a compact support included inRn

+, and note ei for i = 1, . . . , n + m the forward and option payoff functionswe get:

A function f(s) : S → R can be represented as

f(s) = Eν[s(x)], for all s ∈ S,

for some measure ν with compact support, iff for some β > 0:

(i) f(s) is positive semidefinite

(ii) f(eis) is positive semidefinite for i = 1, . . . , n + m

(iii)(

βf(s) −∑n+m

i=1f(eis)

is positive semidefinite

this turns the basket arbitrage problem into a semidefinite program

Semidefinite Programming

A semidefinite program is written:

minimize TrCX

subject to TrAiX = bi, i = 1, . . . , m

X � 0,

in the variable X ∈ Sn, with parameters C, Ai ∈ Sn and bi ∈ R fori = 1, . . . , m. Its dual is given by:

maximize bTλ

subject to C −∑m

i=1λiAi � 0,

in the variable λ ∈ Rm.

Extension of interior point techniques for linear programming show how tosolve these convex programs efficiently (see Nesterov & Nemirovskii (1994),Sturm (1999) and Boyd & Vandenberghe (2004)).

Option Pricing: a Semidefinite Program

We get a relaxation by only sampling the elements of S up to a certaindegree, the variable is then the vector f(s) with

e = (1, x1, . . . , xn, |wT1 x−K1|, . . . , |w

Tmx−Km|, x2

1, x1x2, . . . , |wTmx−Km|N)

testing for the absence of arbitrage is then a semidefinite program:

find f

subject to MN(f(s)) � 0MN(f(ejs)) � 0, for j = 1, . . . , n,

f((β −∑n+m

k=1ek)s)

f(ej) = pj, for j = 1, . . . , n + m and s ∈ S

where MN(f(s))ij = f(sisj) and MN(f(eks))ij = f(eksisj)

Conic Duality

Let Σ ⊂ A(S) be the set of polynomials that are sums of squares ofpolynomials in A(S), and P the set of positive semidefinite sequences on S

• instead of the conic duality between probability measures and positiveportfolios

p(x) ≥ 0 ⇔

p(x)dν ≥ 0, for all measures ν

• we use the duality between positive semidefinite sequences P and sums ofsquares polynomials Σ

p ∈ Σ ⇔ 〈f, p〉 ≥ 0 for all f ∈ P

with p =∑

i qiχsiand f : S → R, where 〈f, p〉 =

i qif(si)

Option Pricing: Caveats

• Size: grows exponentially with the number of assets: no free lunch. . .

• In dimension 2, for spread options, this is:

(k + 1)

where d is the degree of the relaxation and k the number of assets.

• Conditioning issues. . .

Conclusion

• Testing for static arbitrage in option price data is easy in dimension one

• The extension on basket options (swaptions, etc) is NP-hard but goodrelaxations can be found

• We get a computationally friendly set of conditions for the absence ofarbitrage

• Small scale problems are tractable in practice as semidefinite programs

References

Berg, C., Christensen, J. P. R. & Ressel, P. (1984), Harmonic analysis onsemigroups : theory of positive definite and related functions, Vol. 100of Graduate texts in mathematics, Springer-Verlag, New York.

Bertsimas, D. & Popescu, I. (2002), ‘On the relation between option andstock prices: a convex optimization approach’, Operations Research50(2), 358–374.

Black, F. & Scholes, M. (1973), ‘The pricing of options and corporateliabilities’, Journal of Political Economy 81, 637–659.

Boyd, S. & Vandenberghe, L. (2004), Convex Optimization, CambridgeUniversity Press.

Henkin, G. & Shananin, A. (1990), ‘Bernstein theorems and Radontransform, application to the theory of production functions’, American

Mathematical Society: Translation of mathematical monographs81, 189–223.

Nesterov, Y. & Nemirovskii, A. (1994), Interior-point polynomial algorithmsin convex programming, Society for Industrial and Applied Mathematics,Philadelphia.

Sturm, J. F. (1999), ‘Using sedumi 1.0x, a matlab toolbox for optimizationover symmetric cones’, Optimization Methods and Software11, 625–653.

A Harmonic Analysis Solution to the Basket Arbitrage Problemaspremon/PDF/INFORMS05harmonic.pdf · A...

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